60.1.483 problem 496

Internal problem ID [10497]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 496
Date solved : Sunday, March 30, 2025 at 05:20:17 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} \left (y-x \right )^{2} \left ({y^{\prime }}^{2}+1\right )-a^{2} \left (y^{\prime }+1\right )^{2}&=0 \end{align*}

✓ Maple. Time used: 0.225 (sec). Leaf size: 135

ode:=(y(x)-x)^2*(1+diff(y(x),x)^2)-a^2*(diff(y(x),x)+1)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= x -\sqrt {2}\, a \\ y &= x +\sqrt {2}\, a \\ y &= x +\operatorname {RootOf}\left (-2 x -\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2}-2 a^{2}+\sqrt {-\textit {\_a}^{4}+2 \textit {\_a}^{2} a^{2}}}{\textit {\_a}^{2}-2 a^{2}}d \textit {\_a} +2 c_1 \right ) \\ y &= x +\operatorname {RootOf}\left (-2 x +\int _{}^{\textit {\_Z}}-\frac {-2 a^{2}+\textit {\_a}^{2}-\sqrt {-\textit {\_a}^{4}+2 \textit {\_a}^{2} a^{2}}}{\textit {\_a}^{2}-2 a^{2}}d \textit {\_a} +2 c_1 \right ) \\ \end{align*}

✓ Mathematica. Time used: 104.897 (sec). Leaf size: 35180

ode=-(a^2*(1 + D[y[x],x])^2) + (-x + y[x])^2*(1 + D[y[x],x]^2)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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✓ Sympy. Time used: 37.161 (sec). Leaf size: 146

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a**2*(Derivative(y(x), x) + 1)**2 + (-x + y(x))**2*(Derivative(y(x), x)**2 + 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} - x - \int \limits ^{- C_{2} + x} \frac {r^{2}}{- r^{2} - r \sqrt {- r^{2} + 2 a^{2}} + 2 a^{2}}\, dr + \int \limits ^{- C_{2} + x} \frac {r \sqrt {- r^{2} + 2 a^{2}}}{- r^{2} - r \sqrt {- r^{2} + 2 a^{2}} + 2 a^{2}}\, dr, \ y{\left (x \right )} = C_{1} - x - \int \limits ^{- C_{2} + x} \frac {r^{2}}{- r^{2} + r \sqrt {- r^{2} + 2 a^{2}} + 2 a^{2}}\, dr - \int \limits ^{- C_{2} + x} \frac {r \sqrt {- r^{2} + 2 a^{2}}}{- r^{2} + r \sqrt {- r^{2} + 2 a^{2}} + 2 a^{2}}\, dr\right ] \]